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UPSC NDA Math Model Paper 5

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Question : 34 of 120

Marks: +1, -0

If

\log_{6} 27=a

\log_{3} 16=?

$\frac{12-4a}{a}$
$\frac{3-4a}{a}$
$\frac{3-a}{a}$
None of these

Solution:

Given:

\log_{6} 27=a

As we know that,

\log_{a} x^{p}=p \log_{a} x

\Rightarrow 3 \log_{6} 3=a

As we know that

\log_{a} b=\frac{1}{\log_{b} a}

\Rightarrow \frac{3}{\log_{3} 6}=a

\Rightarrow \log_{3} 6=\frac{3}{a}

As we know that,

\log_{a}(x \cdot y)=\log_{a} x+\log_{a} y

\Rightarrow \log_{3}(2 \times 3)=\log_{3} 2+\log_{3} 3=\frac{3}{a}

As we know that,

\log_{a} a=1

\Rightarrow \log_{3} 2=3-\frac{a}{a}

\Rightarrow \log_{3} 16=4 \times \log_{3} 2

Now by substituting the value of

\log_{3} 2

in equation (1), we get

\Rightarrow \log_{3} 16=\frac{12-4a}{a}

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